For the past several years, a rigorous couple wave analysis (RCWA) and similar algorithms have been widely used for the study and design of diffraction structures. In the RCWA approach, the profiles of periodic structures are approximated by a given number of sufficiently thin planar grating slabs. Specifically, RCWA involves three main steps, namely, the Fourier expansion of the electric and magnetic fields inside the grating, calculation of the eigenvalues and eigenvectors of a constant coefficient matrix that characterizes the diffracted signal, and solution of a linear system deduced from the boundary matching conditions. RCWA divides the problem into three distinct spatial regions: 1) the ambient region supporting the incident plane wave field and a summation over all reflected diffracted orders, 2) the grating structure and underlying non-patterned layers in which the wave field is treated as a superposition of modes associated with each diffracted order, and 3) the substrate containing the transmitted wave field.
The accuracy of the RCWA solution depends, in part, on the number of terms retained in the space-harmonic expansion of the wave fields, with conservation of energy being satisfied in general. The number of terms retained is a function of the number of spatial harmonics orders considered during the calculations. Efficient generation of a simulated diffraction signal for a given hypothetical profile involves selection of the optimal set of spatial harmonics orders at each wavelength for both transverse-magnetic (TM) and/or transverse-electric (TE) components of the diffraction signal. Mathematically, the more spatial harmonics orders selected, the more accurate the simulations. However, the higher the number of spatial harmonics orders, the more computation is required for calculating the simulated diffraction signal. Moreover, the computation time is a nonlinear function of the number of orders used. Thus, it is useful to minimize the number of spatial harmonics orders simulated at each wavelength. However, the number of spatial harmonics orders cannot arbitrarily be minimized as this might result in loss of information.
The importance of selecting the appropriate number of spatial harmonics orders increases significantly when three-dimensional structures are considered in comparison to two-dimensional structures. Since the selection of the number of spatial harmonics orders is application specific, efficient approaches for selecting the number of spatial harmonics orders is desirable.